# My Favorite One-Liners: Part 24

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Here’s a problem that could appear in my class in probability or statistics:

Let $f(x) = 3x^2$ be a probability density function for $0 \le x \le 1$. Find $F(x) = P(X \le x)$, the cumulative distribution function of $X$.

A student’s first reaction might be to set up the integral as

$\displaystyle \int_0^x 3x^2 \, dx$

The problem with this set-up, of course, is that the letter $x$ has already been reserved as the right endpoint for this definite integral. Therefore, inside the integral, we should choose any other letter — just not $x$ — as the dummy variable.

Which sets up my one-liner: “In the words of the great philosopher Jean-Luc Picard: Plenty of letters left in the alphabet.”

We then write the integral as something like

$\displaystyle \int_0^x 3t^2 \, dt$

and then get on with the business of finding $F(x)$.